Approximating Routing and Connectivity Problems with Multiple Distances

Abstract

We consider routing and connectivity problems for which the input includes a complete graph G and multiple edge-weight functions d1, d2, ⋯, dr. In each case, a solution is a minimum-cost subgraph H satisfying the constraints of the particular problem. The cost of each edge of H is determined by any chosen function di, but there is a service fee g≥ 0 for each maximal connected component formed by edges associated with the same function. This is motivated by applications for which a solution can be split between multiple providers, each corresponding to a distance di. One example is the Traveling Car Renter Problem (CaRS), which is a generalization of the Traveling Salesman Problem (TSP) whose goal is to visit a set of cities by renting cars from multiple companies. In this paper, we give O(log n) -approximations for the generalizations with multiple distances of several problems (Steiner TSP, Profitable Tour Problem, and Constrained Forest Problem). This factor is the best-possible unless P = NP.